The British prime minister wants a "backstop" measure which would create a temporary "joint customs territory" with the EU for the whole of the UK.

But Brussels appears set to insist on a Northern Ireland-only "backstop to the backstop" in case negotiations on a wider UK approach break down.

In the letter, obtained by The Times, Ms May said: "I am clear that I could not accept there being any circumstances or conditions in which that 'backstop to the backstop', which would break up the UK customs territory, could come in to force."

But the DUP has interpreted the wording of her letter to mean that the measure will be contained in the Brexit divorce deal despite Mrs May's insistence it will never come into effect.

Ms Foster said: "The prime minister's letter raises alarm bells for those who value the integrity of our precious union and for those who want a proper Brexit for the whole of the UK.

"It appears the prime minister is wedded to the idea of a border down the Irish Sea with Northern Ireland in the EU single market regulatory regime."

The DUP's Sammy Wilson speaks to RTÉ radio's Sean O'Rourke

The DUP's Sammy Wilson told BBC Radio 4's Today programme his party has "a number of concerns" and accused Downing Street of leaking the letter sent to DUP leader Arlene Foster, telling the same programme: "I assume it's part of the process of trying to get into the public domain what will finally be agreed."

Mr Wilson also said: "In that letter it makes it quite clear the government has accepted there will be a Northern Ireland-only backstop, that that backstop will require specific alignment for regulations, a solution which does that, and the scope of that will be carefully subscribed.

"Thirdly, the promise which was made in the agreement in December that if there was to be any regulatory alignment then Northern Ireland would have a say on that - that seems to have been removed as well, which would mean regulations which would apply to Northern Ireland would not be decided at Westminster, would have no input from Northern Ireland and would come only from the EU.

The PM’s letter raises alarm bells for those who value the integrity of our precious union & for those who want a proper Brexit for the whole UK. From her letter, it appears the PM is wedded to the idea of a border down the Irish Sea with NI in the EU SM regulatory regime. https://t.co/qN6c41zx7U

"That, to us, is a breach of the promise which has been made that we would not be cut off from the rest of the United Kingdom."

Asked if he accepted Theresa May's pledge of no border in the Irish Sea, Mr Wilson replied: "We want to trust the Prime Minister because she has said so many times that that is the case, but you have to judge any promise by what is actually delivered in an agreement.

"From what we can see in the letter which has been sent to Arlene and Nigel (Dodds), it is quite clear some of the promises made do not conform to some of the content of the letter."

Theresa May's Conservative Party relies on the support of the DUP's 10 MPs for her Commons majority, votes which may become crucial as she attempts to get a deal through Parliament.

A Downing Street spokesman said: "The prime minister's letter sets out her commitment, which she has been absolutely clear about on any number of occasions, to never accepting any circumstances in which the UK is divided into two customs territories.

"The government will not agree anything that brings about a hard border on the island of Ireland."

Any version of the backstop would apply unless and until a wider UK-EU deal on the future relationship solved the issue of how to avoid a hard border with the Republic.

Sinn Féin MLA Máirtín Ó Muilleoir said the British government needed to honour its commitments to avoid a hard border rather than promises made to the DUP.

"The fact is that Theresa May signed up to the Irish backstop in December and that remains the bottom line in order to prevent a hard border and safeguard our political and economic stability now and for the future," he said.

"It cannot be negotiated downwards. It cannot be watered down or bargained off.

“It must be built on to ensure our rights are protected and translated into legally operable and binding text as part of any withdrawal agreement."

Taoiseach Leo Varadkar, Mrs May's effective deputy prime minister David Lidington and Secretary of State Karen Bradley will attend a summit on the Isle of Man on Friday.

Brexit is expected to dominate the agenda of the British Irish Council, which also involves the first ministers of Scotland and Wales, Nicola Sturgeon and Carwyn Jones.

Downing Street has played down suggestions that a Brexit deal is imminent, after European Council president Donald Tusk appeared to indicate a breakthrough could come within the next week.

A senior British government source said that reports in the European media that a deal could come in the next few days should be taken "with a very large pinch of salt".

A potential sticking point could be demands for EU fishing fleets to be given continued access to British coastal waters as the price for agreeing to Mrs May's UK-wide backstop, the Daily Telegraph reported.

A UK-wide customs deal would maintain quota-free and tariff-free access to European markets for the British fishing industry and in return the EU wants to keep continued access to UK waters for its trawlers, the newspaper said.